Understanding Math – It’s All About Perspective

I love to explore TedTalks as there are so many interesting ones that expose you to new ideas. TedTalks are great to use with students as well, because they can spark conversations, provide some real-world applications, and engage students in learning. I am always looking for any math-related TedTalks, especially when I can connect them to concepts students might be about to explore or have already explored and I want to provide an interesting connection.

There was a newer talk posted by Roger Antonsen entitled Math Is the Hidden Secret to Understanding the World where he talks about how mathematics is all about patterns and the idea of finding patterns is how we use mathematics to understand the world. Asking the questions of how does this work, and why does this work.  Representing something with patterns and then changing the perspective of that pattern can lead to really interesting things.  One of my favorite lines of Antonsen’s is “If you change your perspective, and take another point of view, you learn something new about what you are watching, or looking at, or hearing”. He does a great example of looking at a very common equation: x + x = 2x and realizing that this ‘equation’ is actually two different perspectives – one additive, one multiplicative. He goes on to give several examples, and one the whole talk really brought out to me is this idea that if students are allowed to explore and describe and explain their own understanding of patterns in the ways that make sense to them, i.e. their representations, they might have a better understanding of the mathematics themselves. Representing numbers as patterns of pictures or sound – fascinating and engaging. When he looks at fractions from the perspective of music or sound, the ‘sound’ of 4/3 is really beautiful and makes sense. The different perspectives are what allow us to understand the mathematics.

Obviously, the Common Core comes to mind immediately – those Standards of Mathematical Practice that I love! If we look at just a couple things from these practices you can see Antonsen’s idea of changing perspective to understand and make sense of mathematics (and the world):

  • Students look at problems from multiple entry points (i.e perspectives)
  • Students reason abstractly – i.e. abstract what they know and apply it to make sense
  • Students model with mathematics – i.e. use different perspectives to represent something mathematically
  • Students look for and make sense of structure
  • Students look for and express regularity – (patterns)

Common Core practices really speak to Antonsen’s idea of understanding by finding patterns and using different perspectives to make sense of the world. He does a great job of both visually explaining, using mathematics as his example, of how changing perspective helps opens you up to understanding the world and becoming a more empathetic participant in it. It’s all about perspective.

Here is Antonsen’s TedTalk – worth a watch!

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Math, Science, Balloons & Macy’s Thanksgiving Day Parade

ELF ON THE SHELF 2012 Macy'sIt’s that time of year again to do a Thanksgiving themed post. I looked back at what I wrote last year around this time, Engaging in Thanksgiving Data, all about interesting data focused around Thanksgiving (food data for example). Still relevant, if you want to take a look at some of the links to use with students. But obviously I need to do something different for this post.

So, the question then becomes what? If not data about Thanksgiving, what other math/science connections can I make relevant to this particularly U.S.-centric holiday?  The answer – the Macy’s Thanksgiving Day Parade and the large balloons. Thanks to NPR and ScienceFRiday videos/stories, there is already a great video and some interesting math and science that goes on behind and within those large floating balloon animals/creatures that you see in the parade coming up this Thursday.

Did you know? (facts from Francie Diep’s 2014 article in Popular Science, The Science and Engineering of Macy’s Thanksgiving Day Balloons)

  • When staffers come up with a new design for a balloon, they first mold it in clay. Then, to help them calculate how much helium it would take to fill the design, theymacys-parade-2012-pillsbury-doughboy-balloon use good old water displacement.
  • The average balloon requires 12,000 cubic feet of helium. That’s enough to fill about 2,500 bathtubs.
  • The average balloon requires 90 handlers during the parade. Handlers must weigh at least 120 pounds and have no heart, back, or knee problems.
  • The balloons were originally made of rubber. Now, they’re made of fabrics coated in a polyurethane material that’s flexible, durable, and leak-resistant. Polyurethanes are synthetic plastic materials that also commonly show up in couch cushions, insulation, and even in synthetic-fiber clothes.
  • The Pillsbury Dough Boy Balloon pictured above requires 90 handlers, is as high as a 4 story building (so how high is that?), as wide as 7 taxi cabs, and as long as 9 bicycles (so how long??).  And has enough dough to make 4 million crescent rolls….wow!

That’s a lot of math/science going on here: scale modeling, volume, physics, estimation, weight, measurement, and geometry to name a few concepts. Think of the interesting questions that students could ask and then explore! Doing this prior to the actually parade on Thursday would be so much fun, and then having students actually watch the parade and maybe collect some data – i.e. how many balloons, how many people on a specific balloon, what is an average balloon and what would a bigger than average balloon be?

You can even find out about the specific balloons that will be in the parade and other interesting information that you could use to spark some math/science questions prior to the parade. Here are some links that give some facts about balloons and the parade itself:

And finally, here’s a video on the math/science behind the balloons from ScienceFriday

The STEM Around Us

NCTM Innov8, the new team-based conference that NCTM is sponsoring, is going on right now in St. Louis, Missouri. Our team is there of hqdefaultcourse, supporting math teachers with our technology and a great team-building session based on the Wheel of Fortune and the probabilities of winning (session is Friday, November 18 at 10:45 am in Room 265/266). St. Louis brings to mind the very famous St. Louis Gateway Arch, something math teachers attending will probably be exploring and trying to mathematically represent – is it a parabola? (In fact, it is NOT a parabola, but rather a flattened catenary). (Cool 3D mathematical model here).

This idea of looking at real objects and connecting mathematics to them is something math teachers do often. It makes complete sense, and, as I have been teaching a geometry course for Drexel these last several weeks, I have really deepened my appreciation for this idea of looking at our constructed world to find the mathematical connections and relationships. What I think we tend not to do with students, and what we should do much more of, is go beyond the obvious “shape” explorations and function fitting to explore the STEM connections.

What I mean is after we identify the inherent shapes and/or functions in ‘real-world’ objects, start asking questions that get students thinking about the why behind those shapes. The why questions lead to investigation and research by students into science, technology, engineering, and math applications that would take them much deeper into understanding the world around them. And, I wager, this type of questioning will engage students in learning and applying what they learn in a much more relevant and interesting way.  Giving them purpose for learning. And, as a result, we might have more students going into STEM fields.

Some examples:

2016-11-17_15-32-11    download     images

Why, for example, are most buildings polygon shapes, particularly triangles and rectangles? Why don’t we see more circular or cylindrical shapes for buildings, besides the grain silos or water towers? Is there a reason? This is where engineering would come into play – are certain shapes stronger from an engineering perspective?

science-test-tubes-all-the-ladies-in-washington-zsrurj-clipart

 

Why are science and medical tubes cylindrical? Is their a scientific reason for these shapes/containers? Why not use a prism shape, so then you could set the vials down on a table versus having to store them in special holders so they don’t roll away? Is the shape somehow connected to the way molecules or blood cells behave – i.e. science factors that might determine the tools used.  2791136-image-of-the-motherboard-without-a-pc-processor-closeup

Look at all the different shapes on a computer motherboard – there are cylinders, rectangles, squares, networks of curves/lines of wires, prisms…so many things going on. Students could ask whether certain shapes provide better conductivity? Or heat control? How does the height of a component impact it (notice the different heights of the cylindrical components). I don’t even know the questions to ask here, but this is a great example of where technology comes into play.

I feel that if we allowed students to explore beyond simple things like fitting a function to a curve or identifying shapes in a picture, and really focused on STEM applications and reasons behind the use of those specific shapes, we would be encouraging students creativity, curiosity, and developing research capabilities in order to find solutions. It would be so engaging and really get students interested in those STEM careers, but more importantly, a better understanding of the STEM around them.

 

Using Connections to Build Understanding

I am teaching a Geometry & Spatial Reasoning course for Drexel this semester for their math masters program for teachers. Absolutely love it because I am learning so much from my students/peers, but because it really is bringing home the importance of prior knowledge to help build connections and real-world connections in helping students learn versus memorize, and construct and reconstruct based on their ability to make connections.

My students, who are a mix of very new math teachers, experienced teachers, and even some career-switchers still in the early stages of teaching, are having this great discussions on the importance of using prior knowledge to help student make their own connections. Some have been doing this all along, but others, as they themselves struggle with some of the geometric concepts we are ‘learning’ (relearning in some cases), are coming to understand the value in helping students use what they know to build on and connect to new information. Makes it easier to recall, and builds a confidence in students that when faced with an unknown situation/problem, they have the skills and confidence to look at it, break it down or add in things to make the unknown familiar and then look for and make use of structure (see what I did there….Common Core Math Practice #7!) to help reach a solution or develop a new conjecture/conclusion.

As an example, we’ve been doing a lot of work with inscribed angles in circles and how do you help students use prior knowledge to build the idea that an inscribed angle is half the measure of it’s intercepted arc if you don’t want students just memorizing formulas? Basically, the conversations revolve around constantly using prior knowledge to make connections, which might mean you need to add in an auxiliary line to a given shape to ‘see’ something familiar (i.e. a linear pair or a triangle, as examples). A strategy that really helps students look for and make use of the structures they are familiar with to help them make sense of a problem.  Here’s an example of just one way to explore inscribed angles, using previously knowledge about triangles:

inscribed-angle

  • In Fig 1, we have an inscribed angle and its intercepted arc a. How could you show that angle 1 (the inscribed angle) is half the measure of it’s intercepted arc? Here’s where students need to make sense of this structure – what prior knowledge can they use to help them?
  • In Fig 2, they add in a radius (auxiliary line), because they know all radii in this circle (any circle are equal – doesn’t change the original inscribed angle….but now – we have a triangle and a central angle (angle 2).  What do they already know? Well, they know the central angle 2 is the same as the measure of the intercepted arc, which is the same intercepted arc as angle 1 (inscribed angle).
  • In Fig 3, students are looking at the triangle created and using prior knowledge – we can mark the two radii equal, making this triangle an isosceles triangle, which they already know from prior knowledge has two base angles that measure the same (angle 1 & 3). Angle 2 is an exterior angle to the triangle, and angles 1 & 3 are remote interior, which they know from prior knowledge sum to the measure of angle 2. Since angle 1 & 2 are equal (isosceles triangle), that makes them each half of angle 2 (Sum divided by 2). Angle 2 is equal in measure to the intercepted arc, so angles 1 & 2 are each half of that, so the inscribed angle 1 is half the intercepted arc.
  • Fig 4 shows that the relationship holds true even if you change the size of the inscribed angle.

This is of course just one example for an inscribed angle, but they can then use this to show that inscribed angles that are not going through the center of the circle have the same relationships – ie add in auxiliary lines, use linear pairs, or triangles or other known things to help make sense and show new things. Prior knowledge, connections – they really matter.

As teachers, it is our duty to make sure we are modeling and helping students use what they know to build these connections and see the relationships. It takes deliberateness on our part, it requires modeling, it requires setting expectations for students till it becomes a habit (habits of mind) to look for and make sense by pulling in previous knowledge.

Another thing we need to do is make connections to real-world. My students are sometimes struggling with this idea of relating prior knowledge and new ideas to real-world applications, but if you get in the habit, its not so hard to do. Since I am focused on circles and the lines that intersect them now with my class, I pulled up a ready-to-use lesson from Casio’s lesson library that is a great example of a real-world connection to circle concepts that would force the use of previous knowledge.  The lesson is briefly described below:

The Perfect Glass Dome: (here’s a link to the complete, downloadable activity)

  • – Use coordinate geometry to represent and examine the properties of geometric shapes.
  • – Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

This activity uses the Prizm Graphing calculator and picture capability to help build understanding.

dispcap1 The kinds of questions and connections to prior knowledge that can be asked of students just by looking at the image are pretty endless. What relationships do you see (i.e. lots of diameters, or straight angles, lots of central angles, all the angles are 360, are their auxiliary lines we could add to find the areas or relationships or angle measures, etc.).

If you look around, you can probably find a real-world example of most math concepts your are working on with your students. Show them pictures, show them real objects they can get their hands on. Start asking questions. Ask them what they recognize or think they already know. Ask them if they could add something or take away something to see a familiar object/concept. How does that help them? What relationships and connections help them get to something new or interesting?

My Drexel course and student are reemphasizing for me (and them) the importance of prior knowledge to help build connections on a continuous basis, all the time, every day. It helps students think mathematically and consistently use vocabulary and math concepts to deepen and create new understanding and relationships. It also promotes logical reasoning and problem solving – win-win!